A Level Mathematics 9709 Paper 11, Oct/Nov 2024

Identify correct term and obtain $6(kx{)}^2\cdot {\left(\frac{2}{x}\right)}^2$

Equate to 150 and obtain $k=\frac{5}{2}$

Identify correct term $4(kx{)}^3\cdot \left(\frac{2}{x}\right)$ with their value of $k$

Obtain coefficient 125

Differentiate to obtain $2x+a{x}^{-2}$ or equivalent

Equate first derivative to zero, substitute $x=-3$ and attempt value of $a$

Obtain $a=54$

Obtain $b=27$

Use correct sector area formula

Obtain $\frac{1}{2}\times {15}^2\times \frac{2}{5}\pi -\frac{1}{2}\times x^2\times \frac{2}{5}\pi =\frac{209}{5}\pi$ or equivalent

Obtain $[x]=4$

Use correct arc length formula twice

Obtain $22+\frac{38}{5}\pi$

Substitute for $y$ (or $x$) in first equation and simplify

Obtain $10x^2+3kx-40$ [= 0] (or $10y^2+11ky+k^2-360[=0]$)

Attempt $b^2-4ac$ for 3-term quadratic involving $k$

Obtain $9k^2+1600$ (or $81k^2+14400$)

$9k^2+1600>0$

Attempt correct process for solving 3-term quadratic equation in $\sqrt{x}$

Obtain at least $2\sqrt{x}-3=0$ or equivalent

Conclude $x=\frac{9}{4}$ ignore $\frac{9}{16}$

Integrate to obtain form $k_1{x}^2+k_2{x}^{\frac{3}{2}}+k_{3}x$ where $k_1{k}_2{k}_3\mathrm{\ne }0$

Obtain correct $2x^2-2x^{\frac{3}{2}}+x$ or equivalent

Substitute $x=4$ and $y=11$ to attempt value of $c$

Obtain $y=2x^2-2x^{\frac{3}{2}}+x-9$

State or imply centre of $C_1$ is $(-3,5)$

State or imply centre of $C_2$ is $(9,-4)$

Attempt correct process for finding distance between centres

Obtain 15

$R=4$ and $R=8$

Obtain least or greatest distance

Obtain 3 and 27

Differentiate to obtain form $k_1(2x+1{)}^{-\frac{4}{3}}$

Obtain correct $-8(2x+1{)}^{-\frac{4}{3}}$ or unsimplified equivalent

Attempt equation of tangent at $(\frac{7}{2},6)$ with numerical gradient

Obtain $y=-\frac{1}{2}x+\frac{31}{4}$ or equivalent of requested form

Integrate to obtain form $k_2(2x+1{)}^{\frac{2}{3}}$

Obtain correct $9(2x+1{)}^{\frac{2}{3}}$ or unsimplified equivalent

Use correct limits correctly to find area

Obtain 27

Use ${\tan }^2\beta =\frac{{\sin }^2\beta }{{\cos }^2\beta }$

$\cos \beta =-\sqrt{1-a^2}$

Obtain $\frac{a^2}{1-a^2}+3a\sqrt{1-a^2}$

Use correct identity to obtain 3-term quadratic equation in $\sin \theta$

Obtain ${\sin }^2\theta +4\sin \theta +1[=0]$

Attempt to solve quadratic

Obtain 195.5

Obtain 344.5

Differentiate to obtain $5+12x-9x^2$

Attempt to find two critical values by solving quadratic equation or inequality

Obtain values $-\frac{1}{3}$ and $\frac{5}{3}$

Conclude $x<-\frac{1}{3}$, $x>\frac{5}{3}$

Equate first derivative to 9 and simplify to 3 term quadratic

Obtain $x=\frac{2}{3}$

Use $x$-value and corresponding $y$-value to determine value of $k$

Obtain $k=\frac{28}{9}$

State or imply that first 3 terms of GP are $5+d$, $5+4d$, $5+10d$

Form equation $(5+4d{)}^2=(5+d)(5+10d)$ or equivalent

Obtain $d=2.5$

Use correct formula for sum of AP with their value of $d$

Obtain or imply 7700

State or imply GP is $7.5,15,30,\dots$

Use correct formula for sum of GP with their common ratio

Obtain $S_{77}-G_{10}=27.5$

Obtain $b=2$ and $c=\frac{3}{2}$

Obtain $\frac{15}{2}-2{(x-\frac{3}{2})}^2$

State range is $y⩽\frac{15}{2}$ or $\text{f}(x)⩽\frac{15}{2}$ with $⩽$ given or clearly implied (not $<$)

State that reflection is in $x$-axis

State or imply that translation is by $\left(\begin{array}{c}-\frac{3}{2}\\ \frac{15}{2}\end{array}\right)$ or equivalent

Sketch the correct graph appearing in second and third quadrants only

State that each $y$-value is associated with a single $x$-value or equivalent

Sketch the correct graph with suitable labelling to distinguish the two curves

Draw the line $y=x$

Attempt correct process for finding the inverse function

Obtain $\frac{3}{2}-\sqrt{\frac{15}{4}-\frac{1}{2}x}$ or equivalent